Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value

We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation with step-like initial data q(x, 0) = 0 for x ≤ 0 and q(x, 0) = Ae-2iBx for x > 0, where A > 0 and B ∈ ℝ are constants. We show that there are three regions in the half-plane {(x, t){pipe}-∞ < x < ∞, t > 0}, on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x > -4tB, a plane wave region: an elliptic region:. Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method. © 2013 Springer Science+Business Media Dordrecht.